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Growth of groups with incompressible elements, I

Zheng Kuang

TL;DR

"We study a broad class of groups of bounded type arising from tile inflations on simple stationary Bratteli diagrams and develop a non-tree-based method to bound their growth. By encoding tile rules with time-varying automata and introducing the last-moment map, we show that if the set of incompressible elements is finite, the growth satisfies $\gamma_G(R) \preccurlyeq \exp(R^{\alpha})$ for some $\alpha\in(0,1)$, generalizing intermediate-growth phenomena beyond line-like orbital graphs. The paper formalizes the bounded-type framework (including contractions and polynomial growth of orbital graphs), proves the main theorem, and demonstrates the approach with several fragmentation-based examples tied to non-Hausdorff singularities, Penrose tilings, and the Fabrykowski-Gupta group. These results provide a systematic method to obtain subexponential growth bounds for a wide family of non-tree actions, contributing to the broader understanding of intermediate-growth phenomena in groups acting on Cantor-like spaces."

Abstract

We define the class of groups of bounded type from tile inflations. These tile inflations also determine some automata describing the groups. In the case when the automata are stationary, we show that if the set of incompressible elements of a group in this class is finite, then this group has subexponential growth with a bounded power in the exponent. Then we describe some examples with certain special structures of orbital graphs and give explicit ways to find the upper bounds for the growth functions of these groups.

Growth of groups with incompressible elements, I

TL;DR

"We study a broad class of groups of bounded type arising from tile inflations on simple stationary Bratteli diagrams and develop a non-tree-based method to bound their growth. By encoding tile rules with time-varying automata and introducing the last-moment map, we show that if the set of incompressible elements is finite, the growth satisfies for some , generalizing intermediate-growth phenomena beyond line-like orbital graphs. The paper formalizes the bounded-type framework (including contractions and polynomial growth of orbital graphs), proves the main theorem, and demonstrates the approach with several fragmentation-based examples tied to non-Hausdorff singularities, Penrose tilings, and the Fabrykowski-Gupta group. These results provide a systematic method to obtain subexponential growth bounds for a wide family of non-tree actions, contributing to the broader understanding of intermediate-growth phenomena in groups acting on Cantor-like spaces."

Abstract

We define the class of groups of bounded type from tile inflations. These tile inflations also determine some automata describing the groups. In the case when the automata are stationary, we show that if the set of incompressible elements of a group in this class is finite, then this group has subexponential growth with a bounded power in the exponent. Then we describe some examples with certain special structures of orbital graphs and give explicit ways to find the upper bounds for the growth functions of these groups.
Paper Structure (23 sections, 30 theorems, 92 equations, 14 figures, 1 table)

This paper contains 23 sections, 30 theorems, 92 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Sketch of the method
  3. Organization of the paper
  4. Preliminaries
  5. Tile inflations
  6. Time-varying automata describing inverse semigroup actions
  7. Minimal semigroup actions on $\Omega(\mathsf{B})$
  8. Graphs of actions and fragmentations
  9. Various notions of repetitivity of orbital graphs
  10. Subexponential growth estimates via orbital graphs
  11. Bounded type
  12. Definitions of inverse semigroups and groups of bounded type
  13. Stationary case
  14. Contractions
  15. Polynomial growth of orbital graphs
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\mathsf{B}=(V,E,\mathbf{s},\mathbf{r})$ be a simple and stationary Bratteli diagram. Let $G$ be a group of bounded type of homeomorphisms of $\Omega(\mathsf{B})$ described by a stationary automaton $\mathcal{A}$. Suppose $G$ acts minimally on $\Omega(\mathsf{B})$. If the set of incompressible e

Figures (14)

  • Figure 1: Bratteli Diagram $\mathsf{B}$ for binary rooted tree.
  • Figure 2: Bratteli Diagram $\mathsf{B}$ for $(\mathcal{F},\sigma)$.
  • Figure 3: Constructing $\mathcal{T}_{n}$ from $\mathcal{T}_{n-3}$.
  • Figure 4: Bratteli Diagram $\mathsf{B}$ related to Penrose tiling.
  • Figure 5: Continuations of boundary points and locations of connecting points, where $i=1,2$.
  • ...and 9 more figures

Theorems & Definitions (107)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.1
  • Remark 2.2
  • Definition 2.6
  • Definition 2.7
  • ...and 97 more